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On Some Inverse Methods inElectromagneticsT.M. Habashy a & R. Mittra ba Schlumberger-Doll Research Old Quarry RoadRidgefield, CT 06877-4108, USAb Department of Electrical and Computer EngineeringUniversity of Illinois 1406 W. Green St. Urbana, IL61801, USAPublished online: 03 Apr 2012.
To cite this article: T.M. Habashy & R. Mittra (1987) On Some Inverse Methods inElectromagnetics, Journal of Electromagnetic Waves and Applications, 1:1, 25-58, DOI:10.1163/156939387X00081
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On Some Inverse Methods in Electromagnetics
T. M. Habashy
Schlumberger-Doll Research Old Quarry Road Ridgefield, CT 06877-4108, USA
R. Mittra
Department of Electrical and Computer Engineering University of Illinois 1406 W. Green St. Urbana, IL 61801, USA
ABSTRACT
The purpose of this paper is to review some of the inverse methods in electromag- netics for the reconstruction of one-dimensional permittivity and conductivity pro- files using transient or spectral data. Two different categories of inversion schemes, viz., the differential-inverse and integral-inverse algorithms, are discussed, and their relationships to other approaches developed in the fields of quantum me- chanics and geophysics are pointed out.
INTRODUCTION
In this paper we present a brief review of some methods for solving inverse prob- lems in electromagnetics, methods that are useful for the problem of profile in- version of a layered medium whose complex permittivity varies as a function of
depth or the radial distance. The scope of the paper is limited to the discussion of rigorous techniques only, and numerical schemes based on parameter estima- tion algorithms, or approximate 2-D or 3-D inversion methods, e.g. migration techniques and ray methods, are not included.
It is interesting to note that the development of some of the well-recognized inverse methods in electromagnetics can be traced to the progress originally made in a number of other areas in mathematical physics, notably quantum mechan- ics and geophysics. In quantum mechanics, the inverse problem pertains to the
recovery of the potential function in the Schroedinger equation, while the recon- struction of the geological parameters of the earth medium from seismological measurements represents the inverse problem in geophysics. In this paper, we illustrate how some of the inversion techniques, originally developed for the above
problems, also find useful applications in electromagnetics. It should be pointed out, however, that since the electromagnetic waves possess a vector character, in general, the analogy between electromagnetics and some of the other areas in
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physics, where the wave functions are scalar, does not always hold. There exist
exceptions, however, where the vector fields can be decomposed into TE (trans- verse electric) and TM (transverse magnetic) components, each of which satisfy a scalar Helmholtz equation. The inverse problems associated with these equations differ significantly, however, with the TM case being substantially more involved than the TE case.
Many of the publications on the inverse methods have addressed the one- dimensional profile inversion problem, which is governed by a scalar-wave equa- tion, and have restricted their discussions to the TE case. In this paper, we begin by addressing the TE problem, which, as alluded to earlier, is simpler than the TM case. Next, we follow with a brief discussion of the TM case.
The various inversion methods described in this paper can be broadly clas- sified under the categories of the "differential-inverse" or the "integral-inverse" approaches. Typically, methods belonging to the first category are recursive in nature and are sometimes referred to in the literature as "layer-stripping" or
"layer-peeling" approaches. This terminology is appropriately descriptive of the
algorithms employed in the differential-inverse approach, since they reconstruct the medium layer by layer, using a recursive procedure.
The integral-inverse method can be further divided into two subcategories, viz., the Repeated Iterative Approach (Distorted-Born iterative approach) and the exact approaches (Gelfand-Levitan, Marchenko, etc.). We present a review of both of the approaches.
FORMULATION
We limit our analysis to the problem of profile inversion of a one-dimensional
layered medium whose complex permittivity can vary as a function of depth in a planar stratified medium or as a function of radial distance in a cylindrically layered medium. For the sake of simplifying the analysis, we restrict ourselves to the planar stratified case where the medium is assumed to be z -stratified, i.e. the permittivity c and the conductivity o- of the medium are functions of z
only. The source is assumed to be localized, and without any loss of generality, the source characteristics can be chosen such that they excite either a purely transverse electric (TE) or transverse magnetic (TM) type of field.
In the frequency domain, the appropriate wave equations governing the electric field for the TE polarization and the magnetic field for the TM polarization are
where Je and J m are the electric and magnetic current densities, respectively, representing the source and
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and the time convention e-z"t has been used.
Alternatively, in the source-free region, one can represent Maxwell's equations in terms of the transverse components of the electric (ET) and magnetic fields, to obtain the following system of two wave components. For the TE-
polarization, where Ez = 0 , we have
For the TM-polarization, where Hz = 0, the corresponding equations are
where V2 T is the transverse part of the Laplacian operator. In the time domain, assuming that the medium is characterized by a nondis-
persive E(z) and 17(z), (2) and (3) take on the following forms:
TE-polarization:
We use the wave equations (1) to formulate the integral-inverse approaches, whereas (1), (4) and (5) are used in formulating the differential-inverse approaches. For the sake of simplifying the analysis, we assume that the exciting TE source is an electric line current, situated at x = 0, z = z, and extending from y = -oo to y = +cc . In this case, the nonvanishing field components are Hz, Hz and
Ey which are functions of x and z only. For the TM case, we assume the source to be a magnetic line current and, in this case, the nonvanishing field components are Ez, Ez and Hy. I
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THE DIFFERENTIAL-INVERSE APPROACHES
In common with the integral-inverse approach, the layer-peeling procedure in the differential-inverse method can be further subdivided into two distinctly different
algorithms. The first one of these, which is based on the Cholesky algorithm [1-6], also referred to as the downward continuation algorithm [7], is applied to the two-
component wave system comprising the upgoing and the downgoing components. The second approach is directly applied to the wave equation using the method of characteristics [8-11].
Other variations of the fast Cholesky algorithm are the Schur algorithm [1,2], which is the frequency domain counterpart of the fast Cholesky algorithm, and the
dynamic deconvolution method which is based on a Riccati equation derived from the two-component wave system [1,2,12]. These recursive approaches are very fast compared to other exact methods. We limit our discussions to the transient
layer-peeling approaches.
The Method of Characteristics
We assume that the medium is characterized by a nondispersive e(z) and 17( z) . The problem at hand is to simultaneously determine e(z) and 17( z) for z
using the method of characteristics [8-11] which employs transient data that are acquired at the surface z = zm (zs < zi) . For the TE polarization, this requires the knowledge of E, (x, z = and Hz(z, z = Zm, t), which are assumed to be measured for all x > 0 and for all times t > 0. However, time-limited data can still be used to invert a finite domain of the medium [9,11]. We further assume that the temporal dependence of the excitation current is
sufficiently smooth so that the medium is at rest for t < 0. From (la), it follows that the electric field Ey(x, z, t) satisfies the wave equation
subject to the following -boundary conditions on the surface z = zm =
where e(x, t) and h(x, t) are the measured y-component of the electric field and the x -component of the magnetic field, respectively, and
With the specific choice of the temporal dependence of the excitation source, the medium is at rest before the source is turned on at t = 0 . Thus, in addition
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to the above boundary conditions, we have the following conditions derived from
causality
In order to make use of the method of characteristics, we would like to reduce the wave equation (6) to a hyperbolic partial differential equation in two variables. For this purpose, we apply the following weighted Fourier transform, with respect to the variable -c , on all the field components.
The weighting in the Fourier transform, given by the factor elk",l(z-zm), has been introduced to eliminate the term proportional to u in the transformed wave
equation. The reason for introducing this step, is to make the numerical imple- mentation of the inversion scheme more convenient. The absolute value of is taken to prevent u(k,:, z, t) from growing indefinitely as z - +cc .
With this choice of the weighting factor, u(k'Z, z, t) is the Fourier transform of the measured field Ey(x, z, t) at z = Zm. Substituting (11) into (6), (7) and
(10), we obtain the following equation for the wave function u, in the variables
k,,, z and t.
subject to the boundary conditions
together with the following causality conditions
In (13) e and h are given by
and the subscripts on u , and h indicate partial derivatives. For convenience, we next introduce a coordinate variable transformation via
the so-called Liouville transformation. Under this transformation, the coordinate variable z is changed to the travel time r defined by
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Upon integrating (16), we obtain
Note that the travel time r is defined in terms of the distance from which the measurements are carried out at z = z,,, and is measured positively for z > z,,,.
Alternatively, and by representing the wave velocity c(z) as a function of T, we can rewrite (16) as
From (18) we obtain,
Once c(T) is solved for, (19) provides a map for relating the variable T to the coordinate variable z .
Using the transformation given by (18) in (12), (13) and (14), we get
with the boundary conditions
and the causality conditions
There are two characteristic curves for the above wave equation. These are
given by
On these characteristic curves, (20) takes on the following form
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where the operator d in (24) denotes total derivative on a characteristic curve
and is defined by
Figure 1. The two characteristic curves for the wave equation.
In order to solve the inverse problem numerically using (24), one needs to dis- cretize it. To accomplish this, one divides the T - t space into square segments, the side of which has a length A. These segments have to be sufficiently small, so that c and T can be approximated by constants in each segment. Upon dis-
cretizing (24) and keeping track of terms of the order A, we obtain the following equations on the two characteristic curves (see Fig. 1)
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Equation (27) represents a recursive formula that allows one to continue the
data known at the arrival time T to a later arrival time T + A, provided that
C(T), r(T), and T(r) are known. To obtain r(r) and T(T), we set t = T + A in (27) and employ the causality condition which implies that U(T + A, T + A) = 0 = This gives
If the conductivity profile is known, then (31) allows one to determine r( T) and hence c(T + A) from the knowledge of c(T), the data u,(-r,,r + 2A) and
'tlt(T,T + in this case we get
In the above equation, we have used the approximation
Equation (31) also allows one to solve simultaneously for T(T) and r(r) , and hence c(r + A) , from the knowledge of c( T) , the data Ur(T,T + 2A) and
ut(T, T 2.6.) at two distinct wavenumbers, e.g. and In this case, we obtain
In deriving (34a), we have used the approximation given by (33). Thus, the inversion scheme can be described as follows: From the knowledge
of the velocity c(r) and the data ur( T, t) and Ut( T, t) for t > r at two distinct
wavenumbers and obtain c(T+0) and T(T) from (34). Next prop-
agate the data to the next arrival time step at T + A , via (27). Finally, the desired
quantity c(z) is recovered from c(r) using the mapping equation (19). Notice
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that the reconstruction of c( T) and T( T) at a certain arrival time depends on the data reconstructed at previous arrival time steps; hence, the error in retrieving
c( T) and T( T) is accumulative.
Figure 2. Geometrical configuration of the inversion problem for the method of characteristics and the Distorted-Born ap- proach.
An implementation of the method of characteristics [9,11] to the cylindrically stratified medium depicted in Fig. 2, is shown in Fig. 3. In carrying out the
inversion, the profile for the relaxation time was approximated by a stepwise function (in this example it was represented by two loss-layers). From the results shown in Fig. 3, it is clear that the inversion scheme is robust in reconstructing the speed profile, however, it is ill-conditioned in retrieving the loss profile. The reason is that, in this method the speed and loss profiles are obtained from data on the wavefront which represents the early time response and hence, corresponds to the high frequency content of the signal. At high frequencies, the medium acts almost like a lossless medium and hence, the reponse of the medium is less sensitive to the loss in the medium.
Before closing this section we would like to add one final comment with regard to the TM case. In contrast to the TE case, the wave equation governing the
magnetic field in the TM case is a third order partial differential equation, given by
Thus, the above approach is not directly applicable to the TM case.
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Figure 3. The results of the simultaneous inversion of the wave speed and loss profiles using the method of characteristics [9,11]. (a) The speed profile. (b) The two parameter inversion for the loss
profile.
The Cholesky Algorithm
As in the previous section, we assume that the medium is characterized by a
nondispersive e(z) and a(z) . The problem at hand is to determine E(z) for
z > zi in a lossy medium with a known conductivity profile o(z) , using the
Cholesky algorithm [1-5] which employs transient data that are acquired at the surface z = zm < xi . Again, and as in the previous section, the TE polarization case requires the knowledge of Ey(x, z = and z = zm, t), which are
assumed to be measured for all x > 0 and for all times t > 0. However, as in
the case of the method of characteristics, time-limited data can still be used to
invert a finite domain of the medium. Three cases of excitations are considered,
depending on the temporal dependence of the source. The first is an impulse
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excitation, where the leading part of the downgoing signal is an impulse in time. The second is a step excitation, where the downgoing signal exhibits, at most, a
step discontinuity in time. The last excitation is the one that is continuous with
respect to time and causal, which we simply refer to as the causal excitation. From (4), we can show that the transverse components of the electric
and magnetic fields satisfy the following two-
component wave system
To apply the Cholesky algorithm, the above system of equations has to be reduced to a system of coupled, first-order partial differential equations. This can be accomplished via the following series of transformations. First, take a Fourier transform with respect to time with w as the transform variable and then apply another Fourier transform with respect to z with kz the corresponding transform variable. This yields the following system of equations
To reduce (37b) to a first-order partial differential equation, we next substitute = cvp in (38b) and divide the equation throughout by w, thus eliminating one
of the time derivatives. Finally, taking an inverse Fourier transform with respect to time leads to the following set of equations
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and v is the wave velocity, defined by
with the values of p chosen such that Ipl < (¡.to max This last con- dition assures that the wave is propagating for all values of z and if p were chosen such that it did not satisfy the above condition, then there would exist a z = zt > z2 such that p2 , which represents a turning point where the wave would experience total internal reflection. Thus, for z < zt , the medium would be probed by a propagating wave, whereas for z > zt the wave will be evanescent and hence no power would reach this part of the medium. Thus, in this case, one expects that the inversion would not work successfully in inverting e(z) and for z > zt, as it would for z < zt .
It can easily be shown that e(p, z, t) and h(p, z, t), governed by (40), are related to E(x, z, t) and H(x, z, t), that satisfy (37), via a Radon transform as follows:
The Radon transform is, in effect, synthesizing a plane wave response by stack-
ing up those values of E(x, z, t) which would arise from a plane wave emerging at an oblique angle 0 determined from sin 0 = pc,. This is the reason why the Radon transform is sometimes refered to as a slant stack. Notice that the variable
p has the dimension of inverse speed and is, therefore, called the slowness. The next step in deriving the two-component wave system comprising the up-
going and downgoing components is to decompose the field quantities e and h into their upgoing (U) and downgoing (D) components. This is accomplished by introducing the following two transformations. The first of these is the Liouville
transformation, which changes the coordinate variable z into the travel time T defined by
Notice that the T - z map, given by (44), is dependent on the variable p and, hence, different p's yield different T - z maps. This is different from the method of characteristics presented in the previous section, where the map did not depend on the wavenumber.
The second transformation needed to derive the two-component wave system, is a field variable transformation given by
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where 77 = aov is the wave impedance. Upon incorporating the above two trans- formations into (40), we obtain the desired set of equations
where r is a reflectivity function that accounts for the scattering from the medium
inhomogeneities and T is the relaxation time that is related to the absorption inside the medium. These are given by the following expressions
In (46), the subscripts on D and U indicate partial derivatives. As we men- tioned at the beginning of this section, we consider three types of excitations: (a) the impulse excitation, (b) the step excitation, and (c) the causal excitation.
The Impulse Excitation
In this case the leading part of the downgoing wave, which includes the primary field, is an impulse in time. This implies that the leading part of D(r = 0, t) has the form S08(t) , where So is a normalization factor determined by the excitation source. By applying the method of propagation of singularities on (46), it can be shown [1,2] that at a later arrival time T > 0, the upgoing and downgoing parts of the signal will have the following forms:
where u(.) is a unit step function, and D and U are continuous functions in t and T. S(Tr) is a function which is to be determined and which is subject to the boundary condition S(T = 0) = So. Upon substituting (49) into (46) and
applying the method of propagation of singularities, it can be shown that S( T) ,
D(T, t) and U(T, t) are governed by the following equations [1-5]
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Equation (50) has the following solution
As mentioned in the previous section, there are two characteristic curves for the wave equation. On these characteristic curves, (52) take on the following form
, ,
Upon discretizing (54) on the characteristic curves (see Fig. 4), we obtain
If the conductivity profile is known, an inversion algorithm can be devised as follows: From the knowledge of the velocity v( T) and the value of U( T, T+) on the wavefront, one can evaluate r(T) from (51). This allows v(T +.6.) to be obtained from
The next step is to propagate the data from the arrival time T to a later one at T+0 using (55). Notice that this scheme requires the knowledge of v(T --- 0) .
The Step Excitation
In this case, the downgoing wave exhibits, at most, a step discontinuity in time.
Thus, D(T,t) and U(T, t) have the following form
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Figure 4. The recursive patterns for the continuation of: (a) The
downward propagating component D( T, t). (b) The up- ward propagating component U (T, t).
Comparing (57) with (49), it is evident that the corresponding expressions for the
step excitation case can be obtained from those of the impulse excitation case by
setting Sp to zero. This yields
together with (54), which, in a similar way can be discretized into the form given by (55). From (58) and (45b), we get the following expression for v(T)
In addition to (59), we get the following equation in r(T) and T(T) , by setting t = T + 2A in (55b) and using (58)
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Using the approximation given by (56) in (60) gives the following expression for v(T +.6.)
where the conductivity profile is assumed to be known. Thus, the inversion scheme can be outlined as follows: From the knowledge of U(T, t) and D(T, t) for t > T, one can obtain v(T from (61). Using (56) and (55), the data can then be
propagated to r + A . Notice that (59) allows v(T = 0) to be evaluated from the measured data at T = 0, rather than being assumed to be given as in cases (1) (the impulse excitation) and (3) (the causal excitation).
The Causal Excitation
This is the case in which the excitation is sufficiently continuous with respect to time, so that the signal is continuous with respect to time and causal, i.e.
In this case U(T, t) and D(T, t) will satisfy the same equations satisfied by
U(T, t) and D(T,t) of the previous two cases, viz., (54). Upon discretization,
(54) has the form given by (55) which is repeated below for convenience
Upon setting t = T + 2A in (63b) and employing causality ( (62)), we get the
following equation involving T(T) and T(T), which has a form similar to that of
(60):
From (64) it is clear that for this case the inversion can proceed in exactly the same manner as the one outlined in the previous section, the only exception being that it is now necessary to know v at T = 0.
A representative implementation of the Cholesky algorithm [2] is shown in Fig. 5. The inversion was carried out for the scalar wave equation (for the acoustic
case) in a planar stratified medium. In electromagnetics, this corresponds to the TE case. The data used in the inversion is due to an impulse response. In the results shown in Fig. 5, the reconstructed speed profile overlapped on the true
profile that was used in simulating the inversion data. We will close this section with one additional remark pertaining to the dif-
ference between the TE and TM cases. Unlike the TE case, which is governed by (4a,b), (5b) which governs the TM case can not be transformed, using the
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Radon transform, to a first order partial differential equation and, therefore, the
Cholesky algorithm can not be directly employed in the TM case.
Figure 5. Plots of the actual and reconstructed acoustic wave speeds using the Cholesky algorithm [2].
THE INTEGRAL-INVERSE METHODS
We divide the integral-inverse methods into two categories, viz., iterative schemes and exact approaches. The only approximations used in the exact approaches are in the process of implementing these algorithms on the computer. In princi- ple, these numerical approximations can be systematically improved upon to any desired degree of accuracy. In contrast, the iterative approaches are inherently approximate in nature at each iteration step and are difficult to improve upon if they do not converge or if they converge to a solution other than the exact solution. However, the iterative appoaches do possess two salutary features, viz., they have a wider scope of applicability and are conceptually much simpler.
The Distorted-Born Iterative Approach
Introduction
The iterative Born approach has been used primarily for the inversion of either the permittivity or the conductivity profiles of a planar stratified slab of known thickness in a planar stratified medium in connection with the scalar wave equa- tion [13-18]. In [13,14] the algorithm is based on a monochromatic measurement which can be made at a finite number of receivers [14] or at a finite number of incident angles [13,14]. In [15,16] the approach requires a multifrequency measure- ment from a normally incident plane wave, whereas in [1 7,18] the measurement is a transient plane wave which is normally incident [17] or obliquely incident [18].
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In [19] the inversion is carried out for the simultaneous inversion of the permit- tivity and conductivity profiles in a cylindrically stratified lossy medium, where the permittivity and conductivity may vary in the radial direction inside an an- nulus of known thickness and the medium is probed by a dipole-type source. The method employed in [20] is based on a first-order Born approximation applied to the source-type integral equation allowing a closed-form solution that requires a
frequency measurement of the reflection coefficient on the continuous line. This iterative inversion scheme is appealing in that it can solve for the permit-
tivity and conductivity profiles simultaneously [19]. In addition, it is conceptually simple, reducing the solution of the nonlinear inverse problem to the solution of a
sequence of linear integral equations. Although it is not very suitable for rapidly varying profiles or media probed by TM polarized waves (as shown later), it works
reasonably well for smooth profiles with relatively high contrasts (the tolerable contrast level depends on the slab thickness relative to the operating wavelength). Moreover, this inversion scheme may be applied with a finite number of receivers at a single frequency, and it displays a reasonably fast convergence.
In this section we review the Distorted-Born approach applied to the case of an unknown planar slab in a planar stratified medium [14]. We limit the dis- cussions to the case of noise-free data. The inversion scheme is formulated using the source-type integral equation (sometimes referred to in the literature as the volume-current integral equation) employing either transient data or monochro- matic data, i.e. data at a single frequency that are obtained at a finite number of receiver locations (or, alternatively, at a finite number of spectral wavenumbers). Unlike the time domain formulation, the monochromatic approach can easily ac- commodate the case of a lossy as well as a general dispersive medium with no additional complications. In either formulations, the integral equation obtained is then solved using an iterative approach where a Distorted-Born approximation is applied at each iteration step, resulting in rapid convergence. We limit our discussions to the monochromatic case.
Formulation
In formulating the problem using the source-type integral equation, one as- sumes that the unknown inhomogeneous slab (extending from z = zi to z = zo ) is embedded inside a known background medium which is not necessar-
ily homogeneous. The scattering from this inhomogeneous slab is then assumed to be produced by a volume distribution of current induced inside the slab. The field can thus be decomposed into a superposition of two parts. The first part
for TE and H(o) for TM) is produced by the exciting source in the back-
ground medium. The second (E(s) for TE and H(s) for TM) represents the field scattered by the inhomogeneous slab and is given by the field produced by the volume current induced inside the slab. Thus, for the TE polarization we have
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and for the TM polarization, we have
Using the Green's function formulation [21], (66b) and (68b) can be cast in the
following integral form for the TE polarization
For the TM polarization, we get
where and are the Green's functions for the TE and TM polariza- tions, respectively, and are given as the solution of
The source-type integral equation is then solved by applying successive iter- ations where a Distorted-Born approximation is applied at each iteration step. For the iteration scheme to converge, and have to be small. It
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is clear that the term on the right-hand side of (70) and the first term on the
right-hand side of (71) can be made arbitrarily small by either lowering the fre-
quency of operation ko or making small. However, the second term on the
right-hand side of (71) can only be made small by making small, i.e. if is sufficiently close to fc. Thus, one can conclude that by choosing a frequency of operation that is sufficiently low, the iteration can be made to converge in the case of the TE polarization even for fairly large values of whereas in the case of the TM polarization, the only way the iteration will converge is when the
initial guessed profile is chosen to be very close to the true profile fc. Another difference between the TE and TM polarizations lies in the degree of
nonlinearity associated with each case. It is clear that (70) and (71) are nonlinear
integral equations in the unknown profile due to the nonlinear dependence of the field (Ey or Hy) on fc. Using a Born-type approximation in carrying out the iteration is in effect linearizing the problem. This linearization is expected to work better in the case of the TE polarization than in the TM case because the TM case is highly nonlinear compared to the TE case. This can be seen
by transforming (1) into a canonical form for comparison. This is done by first
carrying out a Fourier transform along the x -direction (assuming kz to be the transform variable) and then introducing the field transformations
This yields the Schroedinger-type equation in the field variable u
Note that the right-hand side of (74) can be interpreted as a source term
resulting from the medium inhomogeneities in the region z > zi. From (76), it is thus clear that for the TE polarization, ec is directly calculable from q ; however, it is necessary to solve a second-order nonlinear differential equation to obtain E, for the TM case. Thus, the TM case is relatively nonlinear compared to the TE case.
From the above considerations, we limit our discussions in the rest of this section to the inversion of the TE case. In this case and for an electric line
current, we have
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Upon substituting (77) and (78) into (70), and setting z = zm (which is the value of z at which the measurement is carried out), we obtain the integral equation
where, for the sake of brevity, G denotes the Green's function for the TE polar- ization ( GTE ) governed by (72a) and Q denotes .6.f given by the expression of
(69). In (79) we have used the reciprocity principle in replacing z-, x', z')
by
The Inversion Procedure
As mentioned in the previous section, (79) supplies us with a nonlinear integral equation in the unknown profile c,. To make the inversion tractable, we linearize the problem by using the Born approximation to get the Fredholm integral equa- tion of the first kind
In practice, the measurement data are only available in a discrete and finite form. Let the data be collected by an array of N receivers. In this case, (80) can be represented in the form
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An inversion procedure that can be employed, starts with the choice of an
initial guess of the profile and can be outlined in the following four steps [13-19]:
Step 1: Calculate Fi(z) from (81). Step 2: Compute the correction term Q(z) by solving (83).
Step 3: Update the profile by computing e?l?(z) = + Q(z)
Step 4: Repeat the procedure at step (1), with = c(l) .
The above procedure is iterated until the difference between the measured field and the computed field at the different receiver locations (or alternatively at the different wavenumbers) becomes smaller than a certain specified value.
In most practical problems, the elements of the kernel F(z) are linearly inde-
pendent ; therefore, solutions to (83) almost always exist. However, the solutions are not necessarily unique. There are two sources of nonuniqueness. The first source is the incompleteness of data due to the availability of only a finite number of receiver locations causing the problem to be underdetermined since the num- ber of measurements is usually far less than the number of parameters required to characterize a profile. The second source of nonuniqueness comes from the
presence of what is referred to in literature [22-24] as the nonradiating sources. These are the solutions to the equation
The set of all these solutions is called the annihilator of the kernel Fi(z). For a finite collection of Fi(z) , the annihilator is, in fact, of infinite dimension. The ad- dition of an arbitrarily weighted sum of these solutions to Q(z) would still make
Q(z) satisfy (83). Thus, our knowledge of the observables His does not tell us
anything about the parts of Q(z) that belong to the annihilator; therefore, these
parts must be recovered from information other than that contained in Hi. It is common knowledge that in most practical cases the elements of the annihilator are found to be highly oscillatory functions. Hence, by eliminating these elements from the solutions, we are in effect looking at the smoothly varying part of the
solution, i.e. at a filtered version of the solution which corresponds to the low
spatial frequency components of its spectrum. To be able to carry out a better in- version of the class of nonsmoothly varying profiles, two alternatives are available. The first alternative tries to incorporate some a priori information concerning the
profile of interest, thus narrowing the class of solutions. The second introduces additional measurements (an example would be to include measurements at more than one frequency) that will reduce the degree of indeterminacy, and thus im-
prove on the ill-conditioning of the inversion. Otherwise, one may tolerate the
nonuniqueness if the class of admissible solutions still contains useful and decisive information about the unknown profile.
A widely used method in circumventing the inherent nonuniqueness problem associated with the solution of (83) is to use Tikhonov's regularization method [25, 26]. This approach allows one to impose an additional condition (not derived from
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the measurement data, but which might be a priori information) which enables one to select one of the possible solutions. Although this additional condition is
arbitrary (since the measurements give no basis for the choice of this condition), it provides us with an indirect way of selecting smooth profiles. In this scheme, regularization is achieved by minimizing a cost function which may be defined by the functional expression
The first quantity in the expression of the cost function is the square of the error while the second represents a smoothing term. N, is a Lagrangian multiplier which is used as a tuning parameter by which one can arbitrarily weigh the importance of the two quantities in the expression for the cost function.
It can be easily shown that the function Q>(z) which minimizes the functional of (85) satisfies the integro-differen.tial equation
+ P+(z). dz'Qp.(z')P(z') = P+(z). x (86) dz2n x
Thus, it is clear that by minimizing the cost function defined in (85), the ill-
posed Fredholm integral equation of the first kind given in (83) is transformed to a well-posed integro-differential equation of the second kind. Since satisfies an integro-differential equation, it is therefore differentiable and hence
provides an approximation to Q(z) that is smoothly varying. If Q(z) is in the same class of smooth profiles as then this approach will yield reasonable results. Otherwise, we will obtain a smoothed version of Q(z). n = 0
corresponds to the case of minimizing the norm of Q while n = 1 corresponds to
minimizing the slopes of Q , and n = 2 corresponds to minimizing the curvature
of Q. In the last case of n = 2, the boundary conditions usually chosen are
In the initial steps of the iteration process, (83) is an approximate equation (since it is a result of the Born approximation); hence, more weight (corresponding to larger Et ) should be put on minimizing the second term of the cost function. In the case of n = 0 , this resembles the method of steepest descent. As the iteration
progresses, c(o) approaches the true profile fc thus resulting in (83) becoming more accurate. Hence, more weight (corresponding to smaller u) should be put on minimizing the first term of the cost function (this resembles the Newton
method).
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Figure 6. Simultaneous reconstruction of sinusoidally varying per- mittivity and conductivity profiles with a hump using the
I Distorted-Born approach [19]. The contrast is 1:10, the annulus thickness is 3m. (a) Dielectric constant. (b) Loss
tangent.
Figures 6-7, show an implementation of the Distorted-Born approach [19] to the cylindrically stratified geometry depicted in Fig. 2 which was carried out at a frequency of 10 MHz. In these plots, all dimensions are in meters, per-
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mittivities are normalized with respect to eo and the loss tangent is defined as
. The solid lines represent the unknown profile to be reconstructed. The M€Q curves indicated by the triangles represent the result of the first iteration. In Fig.
6, the initial guessed profile is chosen to be a constant equal to the value of
Figure 7. Reconstruction of a step profile. The contrast is 1:40, the
annulus thickness is 30 cm. (a) Dielectric constant. (b) Loss tangent.
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the complex permittivity in the region p > Ro, whereas in Fig. 7, it is chosen to be a ramp connecting between the values of the complex permittivities at the two boundaries of the annulus.
Figure 6 shows the result of the simultaneous reconstruction of sinusoidally varying permittivity and conductivity profiles of an annulus whose thickness is 3 m. The contrast (defined as the ratio between the minimum to maximum value of the permittivity in the annulus) is 1:10. In this case the iteration converged after 10 trials.
Finally, Fig. 7 is the result obtained for a step profile of contrast 1:40 for an annulus of thickness 30 cm. In this case, the inversion takes 26 trials to converge, and as discussed previously, we get a smoothed version of the profile.
Exact Methods
Introduction
The first complete solution of the inverse problem that is based on an exact
integral approach was obtained by Gelfand and Levitan [27-33] for the potential problem in the Schroedinger wave equation. In electromagnetics, the above ap- proach is directly applicable to the case of inversion with a transient plane wave, normally incident on a planar stratified lossless medium. Other variations of this classical integral-inversion approach have been developed by considering special choices of input-output pairs [1]. Generalizations of the Gelfand-Levitan approach to the case of oblique incidence [34-36], dissipative media [37-43], spectral do- main inversion (as opposed to temporal domain) [44-46], simultaneous inversion of more than one parameter [34-36,39,46-48] , cylindrically stratified media [46,47], etc., were all based on deriving a Schroedinger-type equation from the basic wave
equation through a series of transformations, and reconstructing the unknown
potential, which is related to the medium parameters, via the Gelfand-Levitan
procedure. Other inverse methods which are based on an integral equation and are in the same spirit as the Gelfand-Levitan approach are the ones due to Krein
[1,29,33] and Gopinath and Sondhi [49-51]. A review of some of these integral inverse methods and others can be found in the review paper by Newton [52].
In this section we focus our attention on the Gelfand-Levitan and Marchenko
equations with the view to unifying them via a general integral equation formu- lation which is pertinent to the electromagnetic inverse scattering problem. The
approach we used to derive this equation is similar in spirit to the ones in the pa- pers by Balanis [53] and Bruckstein et al. [1] which is based on a time-domain for-
mulation, as opposed to the traditional spectral-domain approach [28-33], which is formulated in the frequency-domain. For a similar approach, see the review
paper by Burridge [49].
Formulation
As we mentioned in the introduction, the Gelfand-Levitan approach [28] is
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directly applicable, in electromagnetics, to the case of inversion with a transient
plane wave, normally incident on a planar stratified lossless medium, provided that the wave equation is converted to the Schroedinger equation. However, by using the transformed eqs. (40), the approach can be readily generalized to the inversion of a planar stratified lossless medium excited by a point dipole source that gives rise to TE polarized waves. In this section, we will limit our discussions to the first case, viz., the normal incident case.
The wave equation can be converted to the Schroedinger equation by apply-
ing a two-step transformation. The first step is the usual coordinate variable transformation given by
The second step, is a field variable transformation, by which the electric field
E(T, t) and the magnetic field H(T, t) are transformed to the equidimensional quantities e(T,t) and h(T, t) , respectively, as follows
where 77 = is the wave impedance. Upon applying the above mentioned
transformations, it can be shown that the normalized electric and magnetic fields are governed by the following equations
The wave equation is thus transformed to the following Schroedinger equation
subject to the boundary conditions
and the initial conditions
where the second part of (95) follows from causality. The prime in (94b) denotes a derivative with respect to the argument.
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In (93), V is the potential of the Schroedinger equation and is related to the wave speed c( T) by the following expression
From the above equation, it is clear that in order for the mapping from q(T) to the potential V (T) to exist, c(T) has to be twice continuously differentiable.
Once we invert for V(r), q(r) (and consequently c(T)) can be obtained by solving the following differential equation
subject to the following boundary conditions [54]
Notice that from (98b) and (91a), the boundary condition given by (94b) cor-
responds to the following boundary condition on the normalized magnetic field
Thus, R(t) represents the reflected electric field while r(t) corresponds to the reflected magnetic field.
To develop the Gelfand-Levitan-Marchenko equation, we define the following two non-causal impulse responses el(T, t) and e2(T, t) which are governed by (93) and are subject to the following boundary conditions
The above two responses represent two linearily independent solutions of eq.
(93) each of which has a support over 0 < :5 T. If V(T) is sufficiently smooth, then
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where Ë1(T,t) and Ë2(T,t) are bounded in It < T and zero elsewhere. Using time reversal arguments (from the losslessness condition) it can be shown that
On substituting (102) and (103) into (93) and using the method of propagation of singularities, it can be shown that
where the total derivative in (105) is along the characteristic curve
whereas in (106) the total derivative is along the characteristic curve
As we mentioned above, ei(T,t) and e2(r, t) are two linearily independent solutions of (93), hence, the electric field e(T,t) governed by (93), (94) and (95) can, thus, be represented as a linear superposition of these two solutions, as follows
where A(t) and B(t) are arbitrary functions of time to be determined from the
boundary conditions. Upon substituting the boundary conditions given in (94) we obtain
By applying the causality condition of (95), we finally get
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where we have used (102) and (103) and the symmetry property given by (104) and also the fact that Ë1(T,t) has a support over [t[ < T.
Equation (112) allows one to solve for El(T,t) from the knowledge of R(t) and r(t), which then permits one to solve for V(T) from (105). In the next two subsections we show that the Gelfand-Levitan and the Marchenko equations are
special cases of the above integral equation with the imposition of two different
boundary conditions. Thus, it appears that the integral equation (112) is more
general than the ones that have appeared in the literature for problems that are solvable by the Gelfand-Levitan or the Marchenko approach.
The Gelfand-Levitan Integral Equation
To derive the Gelfand-Levitan equation, we let
The corresponding boundary conditions are given by
This is equivalent to a medium probed by a source backed by a perfectly re-
flecting magnetic conductor at T = 0 (since, h(T = O,t) = 0 for t # 0 ). Under the above boundary conditions, the desired potential function V(T) is obtained from the following equation
where K(T, t) is a symmetric kernel given by
and is governed by the following integral equation
Equation (117) was derived by adding (112) to the one obtained from (112) upon replacing t by -t. In deriving the above equation, we also used the causality condition on the reflection coefficient given by
The Marchenko Integral Equation
In this case, we let
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and hence the corresponding boundary conditions are given by
This is equivalent to a medium (occupying T > 0) probed with a right-going plane wave (i.e. propagating along the positive T -axis) in an unbounded half-
space (occupying r < 0). Under the above boundary conditions, the desired
potential function V(r) is obtained from the following equation
where K(T, t) is given by
and is governed by the following integral equation
CONCLUSIONS
In this paper we have presented a brief review of some methods for solving the inverse problem in electromagnetics, methods that are useful for the problem of profile inversion of a layered medium whose complex permittivity varies as a function of depth or the radial distance. The scope of the paper was limited to the discussion of rigorous techniques only.
The various inversion methods described in this paper were broadly classi- fied under the categories of the "differential-inverse" or the "integral-inverse" ap- proaches. Typically, methods belonging to the first category are recursive in
nature, since they reconstruct the medium layer by layer. The integral-inverse method can be further divided into two subcategories, viz., the Repeated Iterative
Approach (Distorted-Born iterative approach) and the Gelfand-Levitan approach. In the course of reviewing these various methods, we have seen that unlike
monochromatic inversion schemes, transient inversion schemes are not easily gen- eralizable to general dispersive media. However, they can handle band-limited data in a much better way than monochromatic ones.
These transient inversion schemes are, in general, robust in inverting for the
speed profile c(r) giving unique solution, but do not provide a robust inversion for
conductivity. On the other hand, inversion schemes based on the monochromatic
source-type integral equation are nonunique, but by supplying a priori informa-
tion, give a robust inversion for both permittivity and conductivity.
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Finally, the approach based on the source-type integral equation is the only one that can be readily generalized to higher-dimension profiles. All other inversion
schemes, described in this paper, apply only to one-dimensional problems where the medium is stratified along one direction.
ACKNOWLEDGMENTS
The Authors would like to thank Drs. A. Sezginer, W. Chew, E. Chow, and K.
Safinya for many helpful and fruitful discussions. The Editor thanks A. J. Devaney, B. C. Levy, H. E. Moses, and one anonymous
Reviewer for their assistance in reviewing the paper.
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Tarek M. Habashy was born in Cairo, Egypt. He received the B.Sc. degree with first degree honors from Cairo University, Cairo, Egypt and the M.Sc. and Ph.D. degrees from the Massachussetts Institute of Technology in 1976, 1980 and 1983, respectively, all in electrical engineering. Since September 1983, he has been with Schlumberger- Doll Research, Ridgefield, Connecticut, as a member of the professional staff conducting research on inverse scattering problems and electromagnetic well-logging techniques.
Raj Mittra is the Director of the Electromagnetic Communication Laboratory of the Electrical and Computer Engineering Department and Research Professor of the Coordi- nated Science Laboratory at the University of Illinois. He is a Fellow of the IEEE and a Past-President of AP-S. He serves as a consultant to several industrial and governmental organizations in the United States and abroad.
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